Dottorato di Ricerca in Fisica
XIX ciclo
Alma Mater Studiorum
Università degli Studi di Bologna
SETTORE DISCIPLINARE: FIS/03TESI PER IL CONSEGUIMENTO DEL TITOLO DI DOTTORE DI RICERCA
PHOSPHORUS ION IMPLANTATION IN
SiC: INFLUENCE OF THE ANNEALING
CONDITIONS ON DOPANT ACTIVATION
AND DEFECTS
Candidata:
Mariaconcetta Canino
Supervisore:
Chiar.mo Prof. Anna Cavallini
Correlatore: Dott. Antonella Poggi
Coordinatore:
Chiar.mo Prof. Fabio Ortolani
Introduction
Semiconductor technology is presently dominated by silicon. The need for different semiconductors is only limited to niche applications, where the use of silicon is prevented. Such fields include high temperature operation, high power and high frequency applications, and optical emission and detection in the UV spectrum. In these fields the development and commercialization of WBG semiconductor- based devices would represent a revolution. On the other hand, switching form silicon technology to a WBG technology is still a challenge because the availability of good substrate material, the fabrication of low-resistance ohmic contacts, and the growth of good quality oxide are still unsolved problems. Furthermore, low-cost process technology, and long term reliability are mandatory for device commercialization.
Among WBG semiconductors SiC represents a promising solution because it is the only one owing a native oxide and because it was thought in the past that SiC processing technology was relatively close to the well-established Si technology. Anyway many obstacles have to be overcome for the development of SiC microelectronics. In the past years much effort was spent in the growth of defect-free material, and recently CREE commercialized zero micropipe wafers (ZMP) [w-cree]. Furthermore the main technological steps for the fabrication of every kind of device, i.e. selective doping and contact manufacturing, need further investigation. The two issues are closely related [Svensson], since, in order to have ohmic contacts with specific contact resistance below 10-5 Ωcm2, doping levels of the order of 1019 cm-3 are required [Choyke].
Problems related with doping by ion implantation are stoichiometric disturbances in the crystal, difficulty in restoring the lattice order, and evaporation and re-deposition of Si resulting in step bunched surfaces. The electrical activation of dopants requires high temperature post implantation annealing, which, on the other hand induces step bunching insurgence [Chen]. Furthermore, even a high temperature annealing is not completely effective in restoring the lattice, thus ion implantation is often performed at high temperature. In case of phosphorus, ion implantation temperatures around 600-700 °C are reported as mandatory [Rao]. The major drawback of performing the implantation at high temperature is that so high temperatures are not feasible in industrial ion implanters.
Step bunching negatively affects device performance. In fact, an increase of the leakage current in the gate of a MOSFET with the gate oxide grown on a step bunched surface was reported in [Capano2001]; the formation of conductive paths among n+/p diodes when a deposited oxide was used as a passivation layer was observed as well [Poggi2002]. Thus, the choice of the annealing temperature must be a compromise between the aim of achieving high electrical activation of dopants and the need to preserve the surface
smoothness. The influence of the heating ramp of the annealing cycle on surface morphology and electrical activation was also reported for Al [Poggi2006] implants, N [Raineri] implants, and N and P [Blanqué] co-implants.
Performing the annealing at low temperature could leave residual implant damage. A high concentration of defects could affect the long term stability of microelectronic devices, thus preventing their commercialization [Advances]. For example the presence of a high concentration of traps can affect the drain current of a MOSFET [Adajaye] resulting in higher threshold voltage and longer switching time [Millman]. Thus, it is important to characterize the electrically active defects in the substrate, and to determine whether they are induced or not by processing, in order to establish a process that does not introduce severe damage in the material.
In this work n+/p diodes were manufactured and characterized by electrical
measurements. A P+ ion implantation at 300 °C is proposed, since this temperature is intermediate between the high temperatures required by SiC technology, and industrial applications. The electrical activation of phosphorus and the effect of the annealing on the surface morphology were analysed for different annealing temperatures and for different heating rates. n+/p diodes were made by combining 300°C P+ ion implantation and the most promising annealing cycle, i.e. 1300 °C for 20 minutes in argon, with a heating ramp equal to 40 °C/s. DLTS analyses were performed on the diodes and on Schottky diodes in order to study the electrically active defects in the material, either native or induced by processing.
A background to SiC is given in chapter 1. Chapter 2 describes the main issues of SiC ion implantation and annealing technology. In chapter three the process for device realization is described. Chapter 4 illustrates the experimental techniques employed in this thesis, i.e. sheet resistance measurements on Van der Pauw geometries, Scanning Capacitance Microscopy (SCM), current voltage (I-V), capacitance voltage (C-V), Deep Level Transient Spectroscopy (DLTS), and Secondary Electron Microscopy (SEM) in Electron Beam Induced Current (EBIC) mode. Results for the electrical activation of the implanted phosphorus, for the electrical performance of the diodes, and for the analyses of electrically active defects induced by the ion implantation are reported in chapter 5. Finally, conclusions are drawn.
1
Summary
1. Silicon carbide1.1 Lattice structure p. 1 1.2 Defects p. 3 1.3 Band structure and physical properties p. 4
2. Ion implantation technology in SiC
2.1 ion implantation p. 9 2.1.1 Physics of ion penetration in solids p. 9 2.1.2 Ion distribution p. 10 2.1.3 Simulation of ion implantation profiles p. 12 2.2 Recovery of the ion implantation damage p. 19 2.2.1 Hot ion implantation p. 19 2.2.2 Annealing
p. 20
3. Fabrication of n+/p junctions in SiC
3.1 Starting material p. 23 3.2 Photolitographic steps p. 23 3.3 Fabrication of n+/p diodes p. 25 3.3.1 Alignment marks p. 25 3.3.2 Ion implantation p. 25 3.3.3 Annealing p. 26 3.3.4 Contacts p. 27
4. Electrical characterizations: theory
4.1 Characterization of electrical properties of doped layers p. 29 4.1.1 Sheet resistance p. 29 4.1.2 Sheet resistance measurements on Van der Pauw geometries p. 30 4.1.3 Scanning Capacitance Microscopy p. 31 4.2 Electrical characterization of a p-n junction p. 31 4.2.1 Properties of p-n junctions p. 31 4.2.2 Current voltage measurements p. 32 4.2.3 Capacitance voltage measurements p. 39 4.3 Capacitance voltage measurements on Schottky diodes p. 41
4.4 Characterization of electrically active defects in semiconductors: DLTS p. 42 4.4.1 Physics of carrier capture and emission at a trap p. 42 4.4.2 Majority carrier trap spectroscopy by DLTS p. 45
4.5 Scanning Electron Microscopy p. 48
4.5.1 Beam-sample interaction p. 48
4.5.2 Images by secondary electrons p. 50 4.5.3 Electron Beam Induced Current p. 50
5. Electrical characterizations
5.1 Experimental setup p. 53 5.2 Devices and test geometries p. 54 5.3 Dopant activation p. 55 5.4 Electrical characterizations of the diodes with the emitter contact in
nickel as deposited p. 59
5.4.1 Current voltage measurements p. 59 5.4.2 Capacitance voltage measurements p. 63 5.5 Electrical characterizations of the diodes with the emitter contact in
nickel silicide p. 66 5.5.1 Effect of nickel silicide on the contact resistance
p. 66
5.5.2 Current voltage measurements p. 68 5.5.3 Capacitance voltage measurements p. 73 5.6 Defects introduced by the ion implantation process p. 73
Conclusions p.83
Chapter 1
Silicon Carbide
1.1 Lattice
structure
Silicon carbide is made up of equal parts of silicon and carbon atoms. The unit cell is tetragonal, consisting of a carbon (silicon) atom, surrounded by four silicon (carbon) atoms at the vertices (fig. 1.1).
Figure 1.1. The tetragonal cell of SiC. The distance between two nearest neighbours is approximately 3.08Å (a) whereas the distance between a C atom and a Si atom is approximately 2.52 Å (b).
The crystal planes are made up of a silicon atom and a carbon atom in the center and on the vertices of a hexagon. This structure is labeled A, and the upper plane can be oriented like B or C as explained in fig. 1.2.
(a) (b)
SiC exists in more than 300 crystal structures, the so-called polytypes, which differ for the stacking sequence of the crystal planes. The stacking sequence of the crystal planes is codified through the Ramsdell notation: each sequence is labeled by a number, which represents the number of crystal planes in the periodic structure, and a letter, H, C, or R, indicating the lattice structure: H means hexagonal, C cubic, and R rombohedral. In fig. 1.3 the stacking sequences of the 3C-, 4H-, and 6H- polytypes are shown. Among SiC polytypes, these are the most studied for device applications.
Figure 1.3. The stacking sequences of 3C-, 4H-, and 6H-SiC.
Each site can be defined as hexagonal or cubic depending on the symmetry of the neighbouring atoms (fig. 1.4). This inequivalence of lattice sites leads also to different energy levels of substitutional impurities, whether they occupy hexagonal or cubic sites.
1.2 Defects
A defect is a portion of material featured by deviations from the ideal lattice structure. We can identify point defects, 2- dimensional defects (dislocations), and 3-dimensional defects.
Point defects can be vacancies, i.e. the absence of an atom at its lattice site, interstitial, i.e. the presence of an atom outside in a site that should be unoccupied. In biatomic crystals, like SiC, antisites, i.e. a Si (C) atom in the C (Si) sublattice, can also exist. Impurities can be observed when an atom of a different species is present in the lattice, either in substitutional or in interstitial position.
Line defects, also called dislocations, are observed when the periodicity of the lattice is broken along a line. Two types of dislocations exist: edge and screw dislocations (fig. 1.5). Edge dislocations can be schematically considered as the removal of a portion of a lattice plane, screw dislocations as a shift of a crystal plane in a direction parallel to the dislocation line.
The amount of distortion can be expressed through the Burgers vector B. In the case of edge dislocations B is perpendicular to the dislocation line, whereas in the case of screw dislocations B is parallel to the dislocation line and its length corresponds to the step height of the dislocation. It has been observed that if B>3c the dislocation forms an open core along the dislocation line. This kind of defect, labeled micropipe, is particularly detrimental for the operation of SiC devices. The progress in crystal growth has brought to a drastic reduction in micropipe density, leading to the commercialization of zero micropipe density [w-cree].
Among 2-dimentional defects that can affect SiC devices, stacking faults are worth a mention. Stacking faults are crystal planes where the stacking sequence of the lattice is broken. In the upper and in the lower regions the lattice is perfect. Stacking faults have been observed to be responsible of the increase of the forward voltage drop of SiC PiN diodes.
Complex defects can be clusters of defects, for example clusters of vacancies, interstitials, or impurities. They can be introduced during epitaxial growth or processing. Each kind of defect has an equilibrium concentration that depends on temperature and on the defect itself:
(a) (b)
Figure 1.5. Representation of edge (a) and screw (b) dislocations. The red symbols indicate the dislocation line.
) / exp( E kT A
NT = − a (1.1). During the growth (at high temperature) a high quantity of impurities can be introduced. After cooling their solubility decreases and as a result precipitates are formed. Processing can induce the formation of clusters of vacancies and interstitials. For example, a heavy ion implantation process creates a high quantity of point defects in the crystal, and after annealing precipitates can be formed [Ohno].
1.3
Band structure and physical properties
Despite technological inconvenience, such as the lack of defect free material and the difficulties still present in processing, SiC is characterized by interesting physical and electrical properties.
The band structure of SiC depends on the polytype. In table 1.1 some properties of SiC, Si [Harris] and other WBG materials [Peréz] are reported.
Table 1.1. Physical and electrical properties of 3C-SiC, 4H-SiC, and 6H-SiC. The properties of Si, GaN and C are reported for comparison.
Material 3C-SiC 4H-SiC 6H-SiC Si GaN C
Band gap Eg(eV) 2.4 3.26 3.03 1.12 3.39 5.45 Intrinsic carrier concentration ni (cm-3) @ 300 K 6.9 8.2×10-9 2.3×10-6 1.5×1010 1.6×10-10 1.6×10-27 Relative dielectric strength ε 9.72 9.66 9.66 11.7 9 5.50 Critical electric field
Ecrit (MV/cm) @ Nd=1017cm-3 2 2.5 2.4 0.6 3.3 5.6 Drift saturation velocity vsat (cm/s) 2.5×107 2×107 2×107 107 2.5×107 2.7×107 Electron mobility μe(cm2/Vs) @ Nd=1016cm-3 750 800 400 1200 1000 1900 Hole mobility μp (cm2/Vs) @ Nd=1016cm-3 40 115 90 420 <850 1600 Thermal conductivity κ (W/cmK) 5.0 4.9 4.9 1.5 1.3 20
Some of these properties vary as a function of temperature and of the doping density.
The band gap amplitude decreases as temperature increases, due to increase of the amplitude of the thermal vibrations of the atoms of the lattice. The interatomic distance increases and the potential in the Schrödinger equation decreases. This results in an enlargement of the bands and thus the value of the band gap decreases. The dependence is given by:
( )
( )
⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + − + + = β β α T T E T Eg g 2 2 300 300 300 (1.2) where α and β are equal to 3.3×10-3 and 0 for 6H- and 4H-SiC [Sze, Sze-2, Bakowski].The intrinsic carrier concentration depends on the temperature and on the band gap: ) exp( 3 2 N N T E kT ni = C V − g (1.3) where NC and NV are the densities of states in the conduction band and in the
valence band, given by:
2 3 2 * 2 2 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = h kT m N e C (1.4a) and 2 3 2 * 2 2 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = h kT m N h V . (1.4b)
Intrinsic carrier concentration is one of the limiting factors of microelectronic devices operation. Wide band gap semiconductors are advantageous because of their capability of operation at higher temperatures than Si. In Si the intrinsic carrier concentration becomes comparable to the doping concentration at about 150°C. In SiC this phenomenon occurs at 900°C. This means that the maximum operating temperature of SiC based devices is not imposed by material properties, but by technological issues, such as contact fabrication or packaging.
In contrast, one of the disadvantages of SiC, and of WBG semiconductors in general, is due to the fact that the electronic levels of the doping species are not as shallow as in Si. This implies that at room temperature only a fraction of the dopants is ionized, i.e. the free carrier concentration, n or p, is lower than the doping concentration.
The fraction of ionised donors ND+ with respect to the total number of
electrically active donors can be calculated by considering the neutrality equation
p N
n= D+ + (1.5) and the Boltzmann relation
⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − = kT E E N n C F C exp , (1.6)
where n is the free carrier concentration, p is the hole concentration, NC is the
density of states in the conduction band, EC -EF the distance of the Fermi level
form the conduction band and T is the temperature.
In a non compensated a n-type material the hole concentration is very small, thus ⎟⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − + − = kT E E g N n F D D exp 1 1 1 1 . (1.7)
The fraction of ionised donors with respect to the total number of electrically active donors is given by
⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − + + − = kT E E N gN kT E E N gN N n D C C D D C C D D 2 exp exp 4 1 1 . (1.8a)
A similar relationship can be obtained for acceptors [Ruff]
⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − + + − = kT E E N gN kT E E N gN N p V A V A V A V A A 2 exp exp 4 1 1 . (1.8b)
This model takes into account only one ionisation level and is valid in the approximation of low compensation and up to concentrations lower than to 1019cm-3 because at higher concentration the degenerate regime is reached and the Boltzmann statistics is no longer valid.
Table 1.2. Energetic levels of Al, B, N, and P, in 4H-, and 6H-SiC. The labels h and k refer to hexagonal and cubic sites, respectively.
SiC polytype E(Al)-EV (meV) E(B)-EV (meV) E(N)-EV (meV) E(P)-EV (meV) 4H 190 300 42 (h) 82 (k) 53 (h) 93 (k) 6H 225 310 82(h) 137 (k) 82 (h) 115 (k)
In table 1.2 the energetic levels of some of the most common doping species are reported for the 4H-, and 6H- polytypes.
The breakdown field in SiC is around 5x higher than in silicon. This is critical for power switching devices, since the specific on-resistance scales inversely as the cube of the breakdown field. Thus, SiC power devices are expected to have specific on-resistances 100 – 150 times lower than comparable silicon devices [Purdue]. The critical electric field is also crucial for high blocking voltage devices (cfr eq. 4.6 ).
SiC devices can operate at high frequencies (RF and microwave) because of the high saturated electron drift velocity of SiC allows high frequency device application, like RF and microwave [w-cree].
The free carrier mobility is limited by the scattering with the fixed ions of the lattice. Since impurities act as scattering centres, the mobility decreases as a function of doping, according to the law [Bakowski, Arora, Schaffer, Schaffer-2]: p n p n T N N N ref p n A D p n p n p n , , 300 1 , , max , , min , . , α γ μ μ μ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + + = (1.9)
where μn,p,min, μn,p,max, αn,p, gn,p, Nn,p,ref are characteristic parameters of the
material. In table 1.3 their values are reported for 4H- and 6H-SiC and for Si. Electrons show higher mobility than holes because of their lower effective mass [w-colorado].
Chapter 2
Ion implantation technology in SiC
The fabrication of devices for commercial use needs the presence of differently doped regions in the material. The only way to achieve selective doping in SiC is ion implantation, due to the extremely high temperatures needed to have diffusion of most dopant impurities in SiC. Since ion beams are employed to modify the chemical or electrical properties of the material, a high amount of damage is created. Lattice recovery takes place during the high temperature annealing treatment performed to electrically activate the dopants. In this section the technological aspects of ion implantation and annealing are examined.
2.1 Ion implantation
2.1.1 Physics of ion penetration in solids
When penetrating into a solid, an ion loses its energy through a series of collisions with the atoms and the electrons of the target. The stopping power of the target material is the spatial rate of energy loss of incident ions, and is due to two contributions, the nuclear stopping power and the electronic stopping power. The first is due to collisions between the incident ion and the nuclei of the target, in which the ion transfers part of its energy to the nuclei of the target, inducing their displacement. This contribution is thus closely related with the implantation damage. The electronic contribution is due to the energy transferred by the ion to the target electrons, which results in excitation or ionization of the target atoms, or excitation of conduction or valence band electrons. Nuclear collisions predominate in case of heavy ions incident with low energy, electronic collisions in case of incidence of very energetic light ions. In fig. 2.1 the dependence of the two interactions on the incident ion velocity is shown.
Since in ion implantation processes of crystalline solids the wafer is oriented in order to avoid channelling of the incident beam, simulations for the ion stopping are made by considering amorphous solids [Wong].
Figure 2.1. Nuclear and electronic stopping power vs incident ion velocity. V0 is the Bohr
velocity (V0= q2/(4πε0ħ) and Z1 is the atomic number of the incident ion.
2.1.2 Ion distribution
The implantation process involves a huge number of events statistical concepts can be used to describe the final distribution of the implanted atoms. The range of an ion is the total distance that the atom travels before coming to rest. Actually the projected range, i.e. the total path length along the direction of incidence, is the quantity of interest. In fig. 2.2 a three-dimensional representation of ion penetration in a solid is shown. The ion enters the solid at the point (0,0,0) at an angle α to the surface normal. It is then stopped at the position (xs,ys,zs). The range is labelled R, the projected range Rp. In the case
of fig. 2.2, since the incident ion is not normal to the surface, we can define also the depth of penetration xs, i.e. the perpendicular distance from the
surface where the ion comes to rest.
The mean projected range is the most probable location for an ion to come to a rest:
∑
= i i p x N R / (2.1) where xi is the projected range of the i-th ion, and N is the total number ofimplanted ions. Rp is also referred to as the first momentum of the ion
distribution.
The average fluctuation, i.e. the second momentum of the ion distribution, (the square root of the standard deviation from the mean projected range) is labelled “straggling”.
(
)
2 12 ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − =∑
N R x i p i σ . (2.2)Figure 2.2. Schematic drawing for the definition of depth, projected range and path length of an incident ion [Eckstein1991].
The third and fourth moments of the ion distribution are calculated as follows:
(
)
( )
∑
− = i p i R N x 3/ σ3 γ (2.3)(
)
( )
∑
− = i p i R N x 4/ σ4 β (2.4) The skewness γ (eq. 2.3) indicates the symmetry of the distribution with respect to the Gaussian distribution: γ>0 indicates that the peak is closer to the surface, γ<0 indicates that the peak is far from the surface. Kurtosis β (eq. 2.4) indicates the extent of the distribution tails: β=3 indicates Gaussian distribution, 0<β<3 indicates abbreviated distribution, and β>3 indicates broad tails.In fig. 2.3 the depth distributions of 4He implanted in Ni at three different energies are shown. The values of the skewness and kurtosis for the three distributions are: 0.1 keV γ=0.8, β=3.4; 10 keV γ=0.1, β=2.5; 1 MeV γ=-4.3, β=42.7.
Figure 2.3 . Depth and spread distributions of He implanted in Ni at 0.1 keV, 10 keV and 1 MeV [Eckstein1991].
2.1.3 Simulation of ion implantation profiles
The accurate simulation of ion implanted profiles is one of the key issues in SiC ion implantation technology.
A calculation of implantation distributions can be made either by running computer programs which perform Monte Carlo simulations of the ion ranges, or by calculating the ion distribution functions using fitting parameters extracted from a database.
The first method makes use of physical models of the ion stopping process. The understanding of the physical mechanisms which regulate the energy loss of ions penetrating in solids is of crucial importance in controlling the depth profile of implanted dopant atoms and in determining the nature of lattice disorder in ion implanted solids as well. A transport calculation was developed in detail by Ziegler and Biersack [Ziegler1985] in the PRAL (Projected Range ALgorithm) code, which is part of the SRIM software package (The Stopping Ions and Ranges in Matter) [w-srim]. This kind of approach allows the calculations of a series of histories, which means that the paths of the ions in the target material are followed. Atom recoils and sputtering are taken into account as well, providing an evaluation of the implantation damage. Another advantage is that the implantation can be simulated for a wide range of targets, including composite materials and layered structures [Ziegler1985, Ziegler1992].
Fig. 2.4 shows 20 ion paths from SRIM Monte Carlo calculations for 100 keV P atoms in SiC, and the ion recoils generated by the nuclear collisions. It is possible to appreciate that the ion paths are irregular, due to the random nature of nuclear scattering.
Figure 2.4. The paths of 20 P atoms implanted at 100 keV into SiC (red lines) and the atom recoils induced by the nuclear collisions of the P ions in the SiC lattice.
Fig. 2.5 shows the ion distribution obtained from the statistics of 15000, 45000 and 90000 ion histories.
0 50 100 150 200 250 -2.0x10-4 0.0 2.0x10-4 4.0x10-4 6.0x10-4 8.0x10-4 1.0x10-3 1.2x10-3 1.4x10-3 1.6x10-3 15000 P ions 45000 P ions 90000 P ions P c onc ent ra ti on ( c m -3 ) Depth (nm)
Figure 2.5. Ion implantation distribution for 100 keV P ions implanted in SiC. The profiles have been obtained with the SRIM Monte Carlo calculation following 15000 (black line), 45000 (red line) and 90000 histories (green line).
Figure 2.6. SRIM output text file for analytically derived stopping power, projected range and straggling of a P ion implanted in SiC.
As one can notice form fig. 2.6, SRIM calculates only the projected range and the straggling. The second and third moments of the distributions are not calculated. Moreover SRIM is time-consuming, since, as one can understand
from fig. 2.5, a wide number of ion histories is needed to generate a profile not affected by noise.
A faster approach allowed by the SRIM software is to calculate the first two moments of the ion distribution as a function of the incident ion energy and of the nuclear and electronic stopping power of the target. As an example a SRIM output file for analytically derived results is given in fig. 2.6 for a 100 keV P ion implanted into SiC. From these parameters one can calculate the ion distribution. There’s a basic difference between the SRIM analytically derived parameters for the ion distribution and the parameters of the distribution extracted by fitting a SIMS profile: the former are derived by using a physical model [Ziegler1985], whereas the latter are calculated from experimental data.
The alternative way of simulating ion implantation profiles is convenient since the profiles generated are as accurate as the experimental data and it is faster once the database has been created. Since in SiC diffusion of species during the annealing does not take place, ion implantation processes are often multiple, i.e. the several ion energies and doses are employed, in order to obtain a flat profile of the implanted species. Thus, a faster simulation approach is convenient in case of SiC because of the complicated implantation schedules required.
The most common distribution functions used for describing ion implantation profiles are the ones of the Pearson family [Ashworth1990]. The ion implantation profiles into crystalline materials in random direction are best described by the Pearson IV distribution:
m p p IV r n A R x r n A R x n M P − ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − + × ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − − = 2 1 arctan exp 1 (2.5) where ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + − = 2 1 2 B r (2.6)
(
2)
12 2 0 4 − − − = ra B B a n (2.7) m = -1/2B2 (2.8) A= mra/n (2.9) a =ΔRp γ (β+3) C (2.10)B Rp2
(
2)
C 0 =−Δ 4β −3γ (2.11) B2 = -(2β-3γ2-6)C (2.12)(
5 6 9)
2 1 2 − − = γ β C . (2.13) The expression 2.5 corresponds to a Pearson IV distribution when 0 < γ2 < 32and β > β0 where β0 is defined by:
(
)
2 5 . 1 2 2 0 32 4 6 39 48 γ γ γ β − + + + = (2.14) There are two ways to calculate a profile from the knowledge of the moments of its distribution. One is to create a database of all extracted moments from which the moments for a desired energy can be extrapolated, the other is to fit appropriate analytical function to the data and tabulate the parameters obtained from the fitting. Janson et al. [Janson2003] chose the second approach. They determined the energy dependence of the first four moments of ion implanted profile distributions by analysing a wide number of SIMS profiles and literature data in 4H- and 6H−SiC.They obtained the following fitting functions for the energy dependence of the distribution moments: ( )2 4 3 2 ln ln 1 r r E a r E a r a r p a E E E R = (2.15) 3 ) / ( 1 2 1 b p E b b R + = Δ (2.16)
(
)
⎟ ⎠ ⎞ ⎜ ⎝ ⎛− − ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − = E c c c E E c c 4 3 4 2 1 exp exp γ (2.17) 0 1 2 5 . 2 β β ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + = d e−d E (2.18)where Er is the energy value (expressed in keV), β0 is defined by eq. 2.14, and
Table 2.2. Least square fit parameters of Eqs. 2.15, 2.16, 2.17, and 2.18 to the distribution moments Rp, ΔRp, γ and β of the experimental and simulated
implantation in Janson’s study [Janson2003]
a1 a2 a3 a4 b1 b2 b3 c1 c2 c3 c4 d1 d2
Ion
(nm) (10-3) (10-3) (nm) (keV) (keV-1/2) (keV) (keV)
1H 16.3 0.76 38.3 7.1 61 5.6 0.99 1.24 1.35 1.4 21 1.4 ∞ 2H 16.4 0.85 38.9 8.1 105 11.4 0.96 0.26 0.44 1.3 151 1.5 241 7Li 3.79 1.12 -27.8 0.3 120 43.4 1.07 1.26 0.44 0.1 245 1.5 10 11B 1.39 1.53 -119 -5.7 124 116 0.76 -0.09 0.06 1.9 83 1.5 519 14N 1.23 1.17 -26.0 0.6 147 297 0.84 -1.21 -0.04 2.8 109 1.5 269 16O 3.47 0.62 72.6 6.5 120 108 1.07 1.62 0.32 0.9 692 1.5 15 27Al 3.34 0.42 113.6 8.4 239 484 0.88 1.76 0.16 2.2 858 1.5 126 31P 1.43 0.74 72.8 6.8 188 411 0.97 0.86 0.02 1.4 476 1.2 ∞ 69Ga 5.60 0.01 122.2 4.1 774 3220 1.11 1.00 0.02 0.0 190 1.1 ∞ 75As 0.84 1.15 -84.6 -7.3 1890 15200 0.87 3.60 0.08 0.7 392 1.1 ∞
0 50 100 150 200 250 -2.0x10-4 0.0 2.0x10-4 4.0x10-4 6.0x10-4 8.0x10-4 1.0x10-3 1.2x10-3 1.4x10-3 1.6x10-3 P c o nc entr at ion ( c m -3 ) Depth (nm) Pearson distribution
SRIM simulation (45000 ions)
Figure 2.7. Distributions of 100 keV P ions in SiC obtained by calculation of the Pearson IV distribution (black line) and by SRIM Monte Carlo calculation (red line).
In this way the distribution of various implanted species can be calculated a-priori. The validity of such calculation has a lower energy limit since the simulated profiles are reliable when the Pearson distribution is not significantly curtailed by the surface [Ashworth1990].
As an example the calculation of the distribution of 100 keV P ions is reported in fig. 2.7. The profile generated by SRIM is reported for comparison.
The differences in the two simulated profiles can be ascribed to some uncertainties that both approaches have. In SRIM the nuclear stopping power is calculated on the basis of a universal potential [Ziegler 1985], obtained by the fitting of experimental values for several targets; the electronic stopping power is a rather complicated function of the incident ion mass, velocity and ionization; the stopping power values obtained for monoatomic targets are tabulated and the values of stopping power for composite targets, like SiC, are obtained analytically by taking into account the stoichiometry and density of the material. In the analytical approach possible error sources can reside in the determination of the SIMS profiles in SiC, in the fitting of the SIMS profiles to obtain the Pearson distribution parameters, and in the fitting of the parameters to obtain their energy dependence. Anyway the position of the peak of the distribution and the straggling yielded by both methods are comparable.
In this thesis the simulation of ion implantation profiles has been achieved with both methods. The data obtained by Janson et al., have been used to create a generator of Pearson distributions.
2.2 Recovery of the ion implantation damage
The ion bombardment leaves some damage in the solid: the formation of primary defects, like vacancies, interstitials, antisites and Frenkel pairs, are the product of the ion cascade. The maximum damage distribution lies closer to the surface than the dopant distribution. It was demonstrated by Monte Carlo calculations that a vacancy rich region is extending closer to the surface, whereas an interstitial rich region is extending between Rp and 2Rp. Moreover,
after implantation the implanted atoms are not electrically active since they occupy interstitial positions. A post implantation thermal treatment is required in order to restore the lattice quality and to electrically activate the dopants [Bentini, Capano].
The main acceptor species in SiC are aluminum and boron. Aluminum is generally preferred because the level that it introduces has lower ionization energy with respect to boron, whereas boron is preferred in case of deep implants because of its lower atomic mass.
The main donor dopants in SiC are nitrogen and phosphorus. They have similar ionization energy. Nitrogen was in former times the favourite choice due to its lower atomic mass, that creates a lower amount of damage. Phosphorus has recently been preferred due to the higher electrical activation that can be achieved even with a low-temperature annealing.
2.2.1 Hot ion implantation
Problems of SiC ion-implantation technology include [Rao]: 1) incongruent evaporation of Si from SiC wafer during post-implantation annealing; 2) stoichiometric disturbances caused by ion-implantation; 3) difficulty in restoring the lattice. These problems result in poor implant electrical activation. Some of these problems are minimized by performing ion implantation at an elevated temperature. When ion implantation is carried out at high temperature defect migration is more likely and crystallization is enhanced [Ziegler 1992]. The crystal quality of a solid can be evaluated by RBS-C. In fig. 2.8 a comparison between the damage introduced by a room temperature P+ ion implantation and by a 800°C implantation is shown after annealing at 1700°C for 30 minutes in argon [Negoro].
The room temperature ion implantation process creates a nearly amorphous layer. Severe damage persists even after a 1700 °C annealing. In the case of hot ion implantation the same annealing at 1700 °C restores the lattice order to the virgin value. This trend is in accordance with the results obtained for other implanted species [Lazar, Inoue]. Phosphorus ion implantation is always performed at temperatures higher than 500-600 °C [Rao]. Since most industrial implanters are not geared for implantation at so high a temperature, it is interesting to study the properties of P doped layer implanted at lower temperature.
Figure 2.8. RBS-C spectra of P+ implanted samples at room temperature and at 800 °C.
2.2.2 Annealing
The post implantation annealing is a necessary step in order to electrically activate the dopants. The annealing is generally performed at temperatures between 1200 °C and 1700 °C. The electrical activation at a certain annealing temperature is characteristic of each species, but the fraction of electrically atoms increasing for increasing annealing temperature for every species. In fig. 2.9 the values of sheet resistance that can be obtained for phosphorus [Capano2000-2] (a) and aluminium [Lazar] (b) ion implantation as a function of the annealing temperature are reported.
It is evident from fig. 2.9 (a) that the implanted phosphorus atoms are electrically activated even by a low-temperature annealing. The drawback of performing the annealing at high temperature is the formation of furrow-like structures. The depth and width of the grooves increases with increasing annealing temperature and time. This effect is referred to as “step bunching”. It is probably due to the off-axis orientation of SiC crystals: during the annealing evaporation of Si species from the crystal surface occurs, and as they re-deposit, they tend to form macrosteps on the cut-off surface [Chen].
(a)
(b)
Figure 2.9. Sheet resistance vs annealing temperature for high dose implanted samples. The data are referred to phosphorus (a) and aluminium (b) implantation.
In order to achieve high donor concentrations phosphorus is a good choice. In fact, it was observed that high concentrations of implanted nitrogen tend to form precipitates, so that the electrical activation undergoes saturation for implanted concentrations higher than 2×1019cm-3. Such behaviour was not
1600 1650 1700 1750 1800 1850 0 10 20 30 40 50 Temperature (°C) R sh (k
Ω
)6H-SiC samples, 30 min annealed 4H-SiC samples, 30 min annealed 6H-SiC samples, 1 hour annealed 4H-SiC samples, 1 hour annealed
observed for phosphorus implants, thus phosphorus is preferred in case of high concentration implants [Rao].
Annealing parameters, such as the annealing temperature, duration, and rate of temperature rise and fall, influence the electrical and morphological features of implanted layers. Effects of the heating ramp on surface morphology and electrically active dopants were reported for aluminium implants [Poggi2006], nitrogen implants [Raineri], and nitrogen and phosphorus co-implants [Blanqué]. The reported data suggest that faster heating rates determine increasing electrical activation, but also increasing step bunching. Fig. 2.10 shows data for this trend in case of aluminium implants and annealing at 1600 °C for 30 minutes in argon [Poggi2006].
(a) (b)
Figure 2.10. Resistivity (a) and average roughness (b) vs annealing heating rate induced by a 1600 °C annealing in aluminium implanted SiC [Poggi 2006].
The decrease of the sheet resistance can be due to: 1) an increase in the electrically active dopant concentration; 2) an increase of the free-carrier mobility due to a reduction of the residual implant damage.
A further effect of the heating ramp is the increase of the reverse leakage current of p+/n diodes with decreasing ramp rate. These results show that the early stages of the annealing cycle play a relevant role on the defects in the implant tail [Poggi2006].
Chapter 3
Fabrication of n
+/p junctions in SiC
The possibility of fabricating n+/p junctions in SiC by lowering the implantation temperature to 300 °C and the annealing temperature to 1300 °C was investigated. The junctions were electrically characterized by current voltage (I–V) and capacitance voltage (C–V) measurements performed on the n+/p diodes. The defects introduced in the p-type layer under the implanted regions were investigated by comparing results of deep level transient spectroscopy (DLTS) measurements carried out both on the n+/p diodes and on Schottky diodes fabricated on the same wafer. The electrical activation of P atoms was evaluated by sheet resisistance measurements on Van der Pauw geometries and TLM structures. An ohmic contact on the n+ regions, made up
of nickel annealed in vacuum at 900°C for 1 minute, was developed. The effect of this thermal treatment on the contact resistance was analysed by TLM measurements, whereas its effect on the series resistance of the diodes was studied by I–V measurements.
3.1 Starting material
A 6H-SiC wafer purchased by CREE research was employed for this study. The substrate was p-type with acceptor concentration 4.2×1018 cm-3, cut 3.5° off-axis. The epilayer was 10 μm thick, with acceptor concentration equal to 7.5×1015 cm-3.
3.2 Photolitographic steps
Junction fabrication in SiC needs a three photolitographic steps process. First, SiC etching for the realization of alignment marks. Second, definition of the implantation regions. Third, definition of the contact areas.
The first photolitography process is necessary because the implanted SiC is to be annealed at very high temperatures, higher than the melting point of whatever mask layer, be it silicon dioxide or metal. Thus, it is necessary to remove the implantation mask layer before annealing. In order to align the contact with the implanted regions the material is first etched and the following masks are aligned on the etched marks.
(a)
(b)
(c)
Figure 3.1. Mask set employed in this study. Alignment mask (a); Implantation mask (b); Metal mask (c). Section views of the samples after the photolithographic steps related to each mask.
The mask layout employed in this study is illustrated in fig. 3.1. In fig. 3.1 (a) the first mask for the alignment marks is shown, and the etched regions are in red. In fig. 3.1 (b) the implantation regions are shown in yellow. In fig. 3.1 (c) the metal mask is shown and the contact areas are in blue. As it can be noticed in fig. 3.1 (b) and (c), Schottky diodes can be fabricated on the p-type epilayer with the same metal used for the ohmic contacts on the n+ regions. Test
3.3 Fabrication of n
+/p diodes
The device fabrication took place in the clean room class 100 of IMM-CNR in Bologna.
First of all the wafer was degreased in boiling acetone and boiling isoporpanol for 5 minutes. Then it was cleaned with a standard Piranha (HCl:H2O2:H2O=1:1:5) for 10 minutes, and with a dip in HF:H2O=1:10 for
30 s.
3.3.1 Alignment marks
The alignment marks were obtained by Reactive Ion Etching (RIE) in SiCl4
plasma. A photoresist film, 1.1 μm thick, was employed as a mask layer. The etching rates for this process were previously calculated to be 50 nm /min for SiC etching and 160 nm/min for photoresist etching. A 5 minute RIE process allowed us to obtain alignment marks 25 nm deep.
The photoresist was then removed by O2 plasma etching, and Piranha cleaning and HF:H2O=1:10 for 30 s were performed.
3.3.2 Ion implantation
The implantation areas were defined in a silicon dioxide mask layer, obtained by low temperature CVD deposition (LTO). The LTO, 600 nm thick, was annealed at 900 °C for 15 minutes in N2 in order to have the same
stoichiometry as a thermal oxide. The implantation areas were defined by using the mask shown in fig. 3.1 (b) .
Table 3.1. Energy and dose values employed in the P+ ion implantation process.
Energy (keV) Dose (cm-2)
10 1.5 1014 20 1.5 1014 40 1.5 1014 60 1.0 1014 100 4.2 1014 170 1.05 1015 Total dose 2.03 1015cm-2
The P+ ion implantation was carried out at 300 °C according to the implantation schedule in Table 3.1. The energy and dose values were calculated by SRIM Monte Carlo simulation in order to obtain a P box profile 200 nm deep with uniform P concentration of 1×1020 cm-3.
In fig. 3.2 the simulated P profile resulting from this implantation schedule is shown. The same profile calculated by constructing the Pearson IV distributions is reported as well.
0 100 200 300 400 500 600 1016 1017 1018 1019 1020 1021 P concentration (cm -3 ) Depth (nm) Simulated P profile SRIM simulation Pearson IV distribution
Figure 3.2. SRIM (back curve) and Pearson IV (red curve) calculations of the P profile resulting from the implantation schedule in table 3.1.
In order to have a low-resistance ohmic contact on the back of the wafer, the substrate p-type doping was reinforced by Al+ implantation. Since for this process a high energy ion implanter was employed, a 400 nm thick SiO2
stopping layer was deposited on the back of the wafer. The Al+ ion implantation was carried out at 400 °C with energy and dose values illustrated in Table 3.2, in order to obtain an Al box profile 280 nm deep, with plateau concentration 8×1019 cm-3.
Table 3.2. Energy e dose values of the Al implantation process on the back surface of the wafer.
Energy (keV) Dose (cm-2)
250 7.2 1014
350 1.5 1015
Total dose 2.22 1015cm-2
3.3.3 Annealing
The electrical activation of the dopants was achieved by annealing at 1300 °C for 20 minutes in a J.I.P.ELEC radiofrequency furnace.
The furnace is equipped with a Ar flux line, and with a vacuum turbo pump which allows pressure values around 10-6 Torr. Several pumping cycles were performed before annealing to avoid oxygen presence in the chamber, and during the annealing a constant Ar flux equal to 150 sccm was provided, in order to have Ar overpressure and avoid oxigen contamination to occur.
The rate of temperature rise was around 40 °C/s, in order to have better P electrical activation, as explained in Chapter 2.
The rate of temperature fall depends on the thermal inertia of the furnace. A graphite susceptor was employed. The temperature in the centre of the susceptor was monitored by a pyrometer. Since RF furnaces are featured by high lateral thermal gradients [Ottaviani], a thick SiC disk was put on the cover of the susceptor, in order to have a better temperature uniformity between the centre and the borders. The wafer was cut into quarters before annealing to have more process variables. The quarters were annealed separately, and held in the centre of the susceptor by two pieces of polycrystalline SiC.
Since a good electrical activation of Al can be achieved by annealing at temperatures around 1600-1700 °C, we expect a partial Al activation. On the other hand, the back surface is completely covered with metal, thus a low contact resistance can be obtained even if the Al activation is low.
3.3.4 Contacts
The back contact was made up of a 80 nm thick Ti layer and a 350 nm thick Al layer, annealed at 1000°C in vacuum for 2 minutes in the J.I.P.ELEC furnace. The back surface of the samples was entirely covered with metal. The pressure during this treatment was between 6×10-6 Torr and 2×10-5 Torr. The ohmic contacts on the n-type regions was made up of Ni. The Ni definition was achieved according to the mask in fig. 3.1 (c), employing the lift-off technique. The lift-off technique can be explained as follows:
1) The sample is covered with photoresist, and the contact geometries are etched in the photoresist
2) The metal is deposited on the photoresist and on the SiC surface, where the photoresist was etched.
3) The photoresist is removed. As a consequence the metal deposited on the photoresist layer in removed as well, whereas the metal on the SiC surface persists.
The lift-off technique was advantageous in this study because of the presence of a metal on the back surface of the samples. Since the front contact definition is achieved by photoresist etching (with a solvent), not metal etching (with and acid), the back metal is not damaged by this treatment. The contact regions were defined in the photoresist, and a Ni layer 50 nm thick was deposited by electron beam.
After this step a first electrical characterisation of the junctions was performed by I-V and C-V measurements. The P electrical activation was evaluated by sheet resistance measurements. DLTS measurements were performed in order to study the electrically active hole traps in the epilayer under the implanted regions.
After the first characterisation the Ni was removed and re-deposited. A thermal treatment in the J.I.P.ELEC furnace at 900 °C for 1 minute in vacuum was performed in order to form NiSi2.
After this alloy treatment the diodes were electrically characterised again (by I-V, C-V and TLM measurements), and DLTS measurements and EBIC analyses were performed.
Chapter 4
Electrical characterizations: theory
4.1 Characterization of electrical properties of doped layers
4.1.1 Sheet resistance
Sheet resistance measurements are carried out in order to determine the concentration of carriers available for conduction in implanted layers.
The sheet resistance, i.e. the quantity that characterizes thin implanted layers, is defined as:
Rsh
t
ρ
= (4.1) Where ρ is the resistivity and t the thickness of the doped layer. It represents
the resistance of a square of side l and thickness t of semiconductor (fig. 4.1):
if a bias is applied to two sides, current flows in the layer, whose section is equal to t×l and whose lenght is l.
Figure 4.1. Schematic representation of a square of side l and thickness t subject to an applied bias.
t
V
I
l
The resistivity is the proportionality factor between the current density and the applied electric field in the layer
E= ρJ . (4.2)
The resistivity is related to the microscopic properties of the material through:
ρ=1/qnμ ; (4.3a) if conduction is due to both carriers
ρ=1/q(neμe+npμp) (4.3b)
where e stands for electrons and p for holes.
From sheet resistance measurements information on the conduction properties of the material can be drawn. The contributions of the mobility and of the concentration of carriers can not be isolated. The carrier mobility is thus generally determined by Hall effect measurements [Soncini].
Due to the incomplete ionization of dopants in SiC, the effective value of the electrically active dopants must be determined by high temperature Hall effect measurement, or Scanning Capacitance Microscopy (SCM) measurements.
4.1.2 Sheet resistance measurements on Van der Pauw geometries
Sheet resistance measurements on Van der Pauw geometries are based on the Van der Pauw theorem [Van der Pauw]: given a flat sample of arbitrary shape and four contacts of arbitrary position along the circumference we define:
AB C D ABCD I V V R = − (4.4) Where VD- VC is the potential difference between the adjacent contacts D and
C and IAB is the current between the adjacent contacts A and B. The quantity
RBCDA is defined similarly. It can be demonstrated that
⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + = BCDA ABCD BCDA ABCD R R f R R t 2 2 ln π ρ (4.5) Where t is the thickness of the layer and f is a function of the ratio RABCD/RBCDA and satisfies the relation
(
)
⎭ ⎬ ⎫ ⎩ ⎨ ⎧ = + − 2 / 2 ln exp arccosh f f R R R R BCDA ABCD BCDA ABCD (4.6)4.1.3 Scanning Capacitance Microscopy measurements
SCM measurements provide information on electrically active dopants in implanted layers. In this technique the tip of an Atomic Force Microscope is biased with a dc signal and an ac signal is superimposed in order to measure a differential capacitance [Blood, w-ois, w-ntmdt].
The bias applied to the tip can be written as:
Vtip=Vdc + Vac sin(ωt). (4.7)
The principle of operation is based on the MOS capacitor physics between the nanometer-sized probe and the underlying semiconductor. A thin oxide is present on the semiconductor surface. The dopant concentration under the probe is related to the change in capacitance, dC, when bias applied to the probe changes the semiconductor surface from accumulation to inversion. dC
is the difference between the accumulation and inversion (or deep-depletion) capacitance. Since a lower dopant concentration results in a lower inversion capacitance, thus a lower dopant concentration will give rise to a larger SCM or dC signal and viceversa [Kopanski].
By scanning the sample surface the map of electrically active dopants is obtained. By scanning the sample cross section the dopant profile is obtained. By comparing SIMS profiles and SCM profiles the fraction of electrically active dopants (i.e. the fraction of the implanted dopants promoted into substitutional position by the annealing) can be evaluated.
4.2 Electrical characterizations of a p-n junction
4.2.1 Properties of p-n junctions
A p-n junction is made up of two regions of semiconductor, one n-type, with donor density Nd, and one p-type, with acceptor density Na. When they are put
in contact a flow of electrons from the n region to the p region takes place, in order to contrast the concentration gradient in the two regions. In proximity of the junctions recombination processes take place and the ionised doping species remain fixed. Equilibrium is reached when the electric field determined by the fixed charge induces a flow of carriers equal and opposite to the one induced by the concentration gradient (fig. 4.2).
The internal potential difference determined by the fixed charge is defined built-in potential, φi. It is equal to the difference between the Fermi energy in
the n-type region and the Fermi energy in the p-type region.
When a negative bias is applied to the p-type region (reverse bias) majority carriers are swept away from the junction, and the band banding increases. This inhibits the current flow (fig. 4.3a). When a positive bias is applied to the p-type region (forward bias) majority carriers are injected into the junction and the band bending decreases. Thus, current flows through the junction (fig. 4.3b). Thus, a p-n junction is rectifying.
Figure 4.3. Band diagram of a p-n junction in reverse (a) and forward (b) bias.
4.2.2 Current-voltage measurements
In this section the mechanisms that rule the current transport in a diodes are described. Current can be due to diffusion of carriers from a region to the other (Shockley equation) and to recombination and generation of carriers in the space charge region due to the presence of electrically active defects.
Diffusion current
The electron current is related to the carrier concentration and to the band banding. n e n e e e n q n kT qn q kT q n q n q kT nE q J μ ⎟⎟= μ −∇ψ + μ ⎢⎣⎡ ∇ψ−∇φ ⎤⎥⎦= μ ∇φ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + ∇ = ( ) ( ) . (4.8)
Similarly for hole current
p p q pn
In thermal equilibrium φn=φp=φ, thus the diffusion current is zero. If a bias is
applied φn and φp are no longer constant all over the junction. The internal
potential difference is given by
V= φp-φn (4.10)
The electron density at the boundary of the depletion layer on the p-side is thus given by ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = kT qV n kT qV p n n p p i p exp 0exp 2 (4.11) Where np0 is the free electron concentration in the neutral p-type region.
Similarly ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = kT qV p pn n0exp . (4.12) From the continuity equation for the steady state we obtain
0 2 2 = ∂ ∂ + ∂ ∂ + ∂ ∂ + − x n D x E n x n E U n n n e n e μ μ (4.13) 0 2 2 = ∂ ∂ + ∂ ∂ − ∂ ∂ − − x p D x E p x p E U n p n p n p μ μ (4.14) 0 / / 2 2 0 = ∂ ∂ + − − ∂ ∂ + − − x p E p n p n x p Da p p n e n p n n n n a n n μ μ τ (4.15) where n n p n n n a n D p D p n D / / + + = (4.16) and U n n U p pn n n n a 0 0 − = − = τ (4.17) are the ambipolar diffusion coefficient and lifetime.
0 2 2 0 = ∂ ∂ − ∂ ∂ + − − x p E x p D p p n p n p p n n μ τ . (4.18)
In the neutral region the electric filed is zero, thus 0 2 2 0 = ∂ ∂ + − − x p D p p n p p n n τ . (4.19) The solution of this equation with the boundary conditions yields
⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = − p n no n n L x x kT qV p p p 0 exp 1 exp (4.20) with p p p D L = τ . (4.21) At x=xn ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = exp 1 kT qV L p qD J p no p p (4.22a) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = exp 1 kT qV L qDen J e po e (4.22b) Thus ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = exp 1 kT qV J J s , (4.23) with e po e p no p s L n qD L p qD J = + (4.24) This is the Shockley equation. It holds in the ideal case, when no traps are
active in the band gap. In case of trap the generation-recombination of carriers at the traps must be considered [Sze].
Generation-recombination current
The generation-recombination current is a due to traps in the band gap [Grove]. Its contribution is more important the higher is the band gap of the material, since it depends on the intrinsic carrier concentration, whereas the diffusion current is proportional to the square of the intrinsic carrier concentration.
The general expression for current in a p-n junction is
qWU
Jgen−rec = (4.25) Where U is the generation rate in the semiconductor [Grove]:
)] exp( [ )] exp( [ ) ( 2 kT E E n p kT E E n n n pn vN U T i i p i T i n i T p n − + + − + − = σ σ σ σ (4.26)
Different approximations are made, according to the bias conditions.
Forward bias
When the junction is forward biased majority carriers are injected: thus, np>ni2 in the whole semiconductor. Though forward bias tends to have band
bending reduced, the presence of a space charge region produces an electric field, which inhibits carrier cross the junction. Thus, current flow is ruled by the recombination of the injected carriers .
The following hypotheses are made: σn=σp=σ np>ni2. Thus, ) exp( ) exp( kT E E n p kT E E n n np vN U T i i i T i T − + + − + = σ . (4.27) Since ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = kT E E n n Fn i iexp (4.28a) ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − = kT E E n p i Fp iexp (4.28b)
and q E E V = Fn− Fp (4.29) thus ) cosh( 2 ) exp( ) exp( ) / exp( 2 kT E E n kT E E n kT E E n kT qV n vN U T i i Fp i i i Fn i i T − + − + − = σ . (4.30)
In the space charge region Ei is halfway between EFn and EFp,
2 Fn Fp i E E E = + . (4.31) The recombination rate is maximum when the recombination centre is near the intrisnsic level, i.e. when the recombination centre is near midgap. The recombination rate becomes therefore
⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = 1 2 exp 2 ) / exp( kT qV kT qV n vN U σ T i (4.32) ) 2 / exp( 2 1 kT qV n vN U = σ T i . (4.33) Since the recombination current is given by:
qWU J = . (4.34) Then ) 2 / exp( 2 1 kT qV n vN qW Jrec = σ T i (4.35) The recombination current depends on the trap concentration and cross section.
Reverse bias
In the case of a reverse biased junction the main process is generation of carriers in the space charge region, because of the reduction in carrier concentration (np<<ni2). Eq. 4.26 becomes:
⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − = kT E Ei kT E E n vN U T i T i T exp exp σ (4.36)
with the hypothesis σn=σp =σ
Also in case of reverse bias U is maximum for a trap near midgap. If we define T T i T vN kT E Ei kT E E σ τ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = exp exp (4.37) then τi n U =− . (4.38) The reverse generation current is thus given by
τi
gen
n qW
J = . (4.39) The reverse current increases with increasing reverse bias due to the dependence
. (4.40)
Thus, taking into account the two contributions, the total current in a diode is thus given by ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = + = kT qV n vN qW kT qV J J J Jf diff rec s T i 2 exp 2 1 1 exp σ (4.41a)
in forward bias, and
τi s rec diff r n qW kT qV J J J J ⎥+ ⎦ ⎤ ⎢ ⎣ ⎡ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = + = exp 1 (4.41b) in reverse bias.
(
V V)
12 W = bi +Fig. 4.4 represents the I-V curve of a diode with respect to the ideal behaviour. In forward bias recombination current is the dominant contribution at low voltages. As the applied bias increases the diffusion current becomes comparable and higher. For this reason the current of a real diode is often expressed by ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = exp ( ) 1 nkT IR V q J J a s s . (4.42)
n is the ideality factor of the diode, and Rs is the series resistance, which takes
into account the contact resistance and the epilayer resistance. The series resistance produces a difference between the applied bias and the effective bias across the junction, but its effect can be neglected at low currents [Sze]. From the I-V curve of a p-n diode the values of Js, n ed Rs can be calculated.
Js is obtained by the extrapolation of the current at zero bias:
J Js =limv→0 (4.43) n is given by J V kT q n ln ∂ ∂ = (4.44) in the exponential region.
And Rs: I V Rs V ∂ ∂ =lim →∞ (4.45) in the high current region.
In reverse bias, in the presence of generating defects, the current increases with increasing bias because of the effect of the dependence of the depletion layer width on the applied bias.
For high values of applied reverse bias, an avalanche multiplication mechanism is generated, and a high current crosses the junction. The threshold values is defined breakdown voltage, Vbd. When the applied bias is
higher than Vbd the carriers accelerated by the high electric field, have enough
energy to ionise the atoms of the lattice. Electron-hole pairs are created and the avalanche mechanism is generated.
If the threshold value for the electric field to have avalanche multiplication of carriers is Ecrit, then the breakdown voltage is related to the epilayer doping
and thickness through the relation [Grove]:
s epy a epy crit bd W qN W E V ε 2 2 − = . (4.46)
